Graphing Sine and Cosine Functions

9.4

Graphing Sine and Cosine Functions

Essential Question What are the characteristics of the graphs of the

sine and cosine functions?

Graphing the Sine Function Work with a partner. a. Complete the table for y = sin x, where x is an angle measure in radians.

x

-2 --- 74 --- 32 --- 54 - --- 34 ---2 ---4 0

y = sin x

x

--4

y = sin x

--2

-- 34

-- 54 -- 32 -- 74 2 -- 94

b. Plot the points (x, y) from part (a). Draw a smooth curve through the points to sketch the graph of y = sin x.

y

1

-2

-32

-

-2

-1

3

2

5 x

2

2

2

c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over -2 x 2. Is the sine function even, odd, or neither?

LOOKING FOR STRUCTURE

To be proficient in math, you need to look closely to discern a pattern or structure.

Graphing the Cosine Function Work with a partner. a. Complete a table for y = cos x using the same values of x as those used in

Exploration 1. b. Plot the points (x, y) from part (a) and sketch the graph of y = cos x. c. Use the graph to identify the x-intercepts, the x-values where the local maximums

and minimums occur, and the intervals for which the function is increasing or decreasing over -2 x 2. Is the cosine function even, odd, or neither?

Communicate Your Answer

3. What are the characteristics of the graphs of the sine and cosine functions? 4. Describe the end behavior of the graph of y = sin x.

Section 9.4 Graphing Sine and Cosine Functions 485

9.4 Lesson

Core Vocabulary

amplitude, p. 486 periodic function, p. 486 cycle, p. 486 period, p. 486 phase shift, p. 488 midline, p. 488

Previous transformations x-intercept

What You Will Learn

Explore characteristics of sine and cosine functions. Stretch and shrink graphs of sine and cosine functions. Translate graphs of sine and cosine functions. Reflect graphs of sine and cosine functions.

Exploring Characteristics of Sine and Cosine Functions

In this lesson, you will learn to graph sine and cosine functions. The graphs of sine and cosine functions are related to the graphs of the parent functions y = sin x and y = cos x, which are shown below.

x

-2 - -- 32 - - --2

0

y = sin x

0

1

0 -1 0

--2

-- 32 2

1 0 -1 0

y = cos x 1

0 -1 0

1

0 -1 0

1

range: -1 y 1

maximum value: 1

y y = sin x

1

-32 - -2 -1

minimum

value: -1

3

2

2

period: 2

amplitude: 1

2 x

maximum value: 1

y = cos x

y

range: -1 y 1

-2 -32 - -2 -1

minimum

value: -1

amplitude: 1

3 2 x

2

2

period: 2

Core Concept

Characteristics of y = sin x and y = cos x ? The domain of each function is all real numbers.

? The range of each function is -1 y 1. So, the minimum value of each function is -1 and the maximum value is 1.

? The amplitude of the graph of each function is one-half of the difference of the maximum value and the minimum value, or --12[1 - (-1)] = 1.

? Each function is periodic, which means that its graph has a repeating pattern. The shortest repeating portion of the graph is called a cycle. The horizontal length of each cycle is called the period. Each graph shown above has a period of 2.

? The x-intercepts for y = sin x occur when x = 0, ?, ?2, ?3, . . ..

? The x-intercepts for y = cos x occur when x = ? --2 , ? -- 32, ? -- 52, ? -- 72, . . ..

486 Chapter 9 Trigonometric Ratios and Functions

REMEMBER

The graph of y = a f(x) is

a vertical stretch or shrink of the graph of y = f (x) by a factor of a.

The graph of y = f (bx) is a horizontal stretch or shrink of the graph of y = f(x) by a factor of --b1.

Stretching and Shrinking Sine and Cosine Functions

The graphs of y = a sin bx and y = a cos bx represent transformations of their parent functions. The value of a indicates a vertical stretch (a > 1) or a vertical shrink (0 < a < 1) and changes the amplitude of the graph. The value of b indicates a horizontal stretch (0 < b < 1) or a horizontal shrink (b > 1) and changes the period of the graph.

y = a sin bx

y = a cos bx

vertical stretch or shrink by a factor of a

horizontal stretch or shrink by a factor of --b1

Core Concept

Amplitude and Period

The amplitude and period of the graphs of y = a sin bx and y = a cos bx, where a and b are nonzero real numbers, are as follows:

Amplitude = a

Period

=

2

-- b

Each graph below shows five key points that partition the interval 0 x -- 2b into four equal parts. You can use these points to sketch the graphs of y = a sin bx and y = a cos bx. The x-intercepts, maximum, and minimum occur at these points.

(y

1 4

2 b

,

a

y = a sin bx

( 2 b

,

0

( (0, 0)

1 2

2 b

,

0

x

(3 4

2 b

,

-a

y

y = a cos bx

(0, a)

(1

4

2 b

,

0

( 2 b

,

a

(

(3

4

2 b

,

0

x

(1

2

2 b

,

-a

(

( (

( (

( (

REMEMBER

A vertical stretch of a

graph does not change its x-intercept(s). So, it makes sense that the x-intercepts of g(x) = 4 sin x and f (x) = sin x are the same.

4

g

f

-4

9 4

-4

Graphing a Sine Function

Identify the amplitude and period of g(x) = 4 sin x. Then graph the function and describe the graph of g as a transformation of the graph of f (x) = sin x.

SOLUTION

The function is of the form g(x) = a sin bx where a = 4 and b = 1. So, the amplitude

is a = 4 and the period is -- 2b = -- 21 = 2.

( ) Intercepts: (0, 0); --21 2, 0 = (, 0); (2, 0)

y 4

( ) ( ) Maximum: --41 2, 4 = --2, 4 ( ) ( ) Minimum: --34 2, -4 = -- 32, -4

3

x

2

2

The graph of g is a vertical stretch by a factor of 4 of the graph of f.

Section 9.4 Graphing Sine and Cosine Functions 487

STUDY TIP

After you have drawn one complete cycle of the graph in Example 2 on the interval 0 x 1, you can extend the graph by repeating the cycle as many times as desired to the left and right of 0 x 1.

REMEMBER

The graph of y = f (x) + k is a vertical translation of the graph of y = f (x). The graph of y = f (x - h) is a horizontal translation of the graph of y = f (x).

Graphing a Cosine Function

Identify the amplitude and period of g(x) = --21 cos 2x. Then graph the function and describe the graph of g as a transformation of the graph of f (x) = cos x.

SOLUTION

The function is of the form amplitude is a = --21 and the

g(x) = a period is

cos -- 2b

bx where a = -- 22 = 1.

=

-- 2 1

and

b

=

2.

So,

the

( ) ( ) ( ) ( ) Intercepts: --14 1, 0 = --14 , 0 ; --43 1, 0 = --43 , 0

( ) ( ) Maximums: 0, --12 ; 1, --21

y 1

( ) ( ) Minimum: --12 1, ---12 = --12, ---21

1

2x

-1

The graph of g factor of -- 21 of

is a the

vertical shrink graph of f.

by

a

factor

of

-- 2 1

and

a

horizontal

shrink

by

a

Monitoring Progress

Help in English and Spanish at

Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of its parent function.

1. g(x) = --14 sin x 2. g(x) = cos 2x

3. g(x) = 2 sin x 4. g(x) = --13 cos --12x

Translating Sine and Cosine Functions

The graphs of y = a sin b(x - h) + k and y = a cos b(x - h) + k represent translations of y = a sin bx and y = a cos bx. The value of k indicates a translation up (k > 0) or down (k < 0). The value of h indicates a translation left (h < 0) or right (h > 0). A horizontal translation of a periodic function is called a phase shift.

Core Concept

Graphing y = a sin b(x - h) + k and y = a cos b(x - h) + k

To graph y = a sin b(x - h) + k or y = a cos b(x - h) + k where a > 0 and b > 0, follow these steps:

Step 1

Identify vertical

the amplitude a, the shift k of the graph.

period

-- 2b,

the

horizontal

shift

h,

and

the

Step 2 Draw the horizontal line y = k, called the midline of the graph.

Step 3 Find the five key points by translating the key points of y = a sin bx or y = a cos bx horizontally h units and vertically k units.

Step 4 Draw the graph through the five translated key points.

488 Chapter 9 Trigonometric Ratios and Functions

LOOKING FOR STRUCTURE

The graph of g is a translation 3 units up of the graph of f (x) = 2 sin 4x. So, add 3 to the y-coordinates of the five key points of f.

LOOKING FOR STRUCTURE

The graph of g is a translation 3 units right of the graph of f(x) = 5 cos --12x. So, add 3 to the x-coordinates of the five key points of f.

Graphing a Vertical Translation

Graph g(x) = 2 sin 4x + 3.

SOLUTION

Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.

Amplitude: a = 2

Horizontal shift: h = 0

Period: -- 2b = -- 24 = --2

Vertical shift: k = 3

Step 2 Draw the midline of the graph, y = 3.

Step 3 Find the five key points.

( ) ( ) ( ) ( ) On y = k: (0, 0 + 3) = (0, 3); --4, 0 + 3 = --4, 3 ; --2, 0 + 3 = --2, 3

( ) ( ) Maximum: --8, 2 + 3 = --8, 5

y

( ) ( ) Minimum: -- 38, -2 + 3 = -- 38, 1

5

Step 4 Draw the graph through the key points.

1

-1

x

4

2

Graphing a Horizontal Translation

Graph g(x) = 5 cos --21 (x - 3).

SOLUTION

Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.

Amplitude: a = 5

Horizontal shift: h = 3

Period:

-- 2b

=

2

--

-- 1 2

=

4

Vertical shift: k = 0

Step 2 Draw the midline of the graph. Because k = 0, the midline is the x-axis.

Step 3 Find the five key points.

On y = k: ( + 3, 0) = (4, 0); (3 + 3, 0) = (6, 0)

Maximums: (0 + 3, 5) = (3, 5); (4 + 3, 5) = (7, 5)

Minimum: (2 + 3, -5) = (5, -5)

y 6

2

-2

x 3 5 7 9

Step 4 Draw the graph through the key points. -6

Monitoring Progress

Help in English and Spanish at

Graph the function. 5. g(x) = cos x + 4

( ) 6. g(x) = --21 sin x - --2

7. g(x) = sin(x + ) - 1

Section 9.4 Graphing Sine and Cosine Functions 489

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